Polarization tensors for massive arbitrary-spin particles and the Behrends–Fronsdal projection operator

Based on the Wigner unitary representations for the covering Poincaré group $ISL(2,\mathbb C)$, we construct spin–tensor wave functions of free massive arbitrary-spin particles satisfying the Dirac–Pauli–Fierz equations. We obtain polarization spin–tensors and indicate conditions that fix the density matrices (Behrends–Fronsdal projection operators), which determine the numerators in the propagators of the fields of such particles. Using such conditions extended to the multidimensional case, we construct a generalization of Behrends–Fronsdal projection operators (for any number $D>2$ of space–time dimensions) corresponding to a symmetric representation of the $D$-dimensional Poincaré group.